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The Mathieu equation (https://en.wikipedia.org/wiki/Mathieu_function) is $y''+(a-2q\cos(2z))y=0$. The modified Mathieu equation is obtained by replacing $z$ with $\pm iz$: $$y''-(a-2q\cosh(2z))y=0.$$ ...

Consider the following Sturm-Liouville problem, $$(\sqrt{\sin \theta} Y')' + \lambda \sqrt{\sin \theta} Y =0$$ where $Y(\theta):[0,\pi] \to \mathbb{R}$ with boundary conditions $Y'(0)=Y'(\pi)=0.$ I ...

I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the ...

Is the eigenvalue decomposition of the Sturm-Liouville operator $$ Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0, $$ with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\...